Question) Reversible vs. Free. Consider the expansion of a system containing pur

Question

Question) Reversible vs. Free. Consider the expansion of a system containing pure methane gas, which can be fairly accurately described by the van der Waals equation of state. Assume that the isochoric heat capacity of methane is 5/2 R and is therefore independent of T.

a. Determine the final temperature, T2, when 1.0 mole of methane undergoes a reversible adiabatic expansion from V1 = 1.0 L to V2 = 20 L, initially at T1 = 298 K.

b. Determine the final temperature, T2, when 1.0 mole of methane undergoes a free adiabatic expansion from V1 = 1.0 L to V2 = 20 L, initially at T1 = 298 K.

c. Describe the qualitative relationship between the heat, work, and U of these two processes (i.e., greater than, less than, or equal)?

Explanation / Answer

For a gas obeying Vanderwall equation of state

(V-b)R(RT)CV = const.

Or (V1-b)R(RT1)CV =(V2-b)R(RT2)CV

for Methane gas a= 2.283 L2.bar/mole2 =2.283*0.9869 L2.atm/mole2 = 2.253 L2.atm/mole2 and b= 0.04278 L/mole, CV= 2.5R

(1-0.04278) 8.314 *( 8.314* 298)2.5R = (20-0.04278)8.314*(8.314*T2)2.5R

1.079*10-11 *1.339*10^49 = T2(2.5*8.314)

T2= 68.5 K

For Free expansion of Vanderwaal gas , deltaT= (2an/3R)*(1/Vf-1/Vi)

=(on, 2*2.253*1)/(3*0.0821 L.atm/mole.K) *(1/20-1/1)=-19.5

T2= 298-19.5 = 278.5 K

In case of adaibatic expansion, the change in internal energy is more for adiabatic expansion than free exoansion.

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